Subgroups — #26214791

Order 12032.

1206 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689235	C₂	1	2	yes	maximal
2	#20245749	C₂	1	2	yes	maximal, complemented
3	#20243287	C₂	1	2	yes	maximal, complemented
4	C₄×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243213	C₂	1	2	yes	maximal, complemented
6	#102×Dic₄₇	C₂	1	4	yes	maximal, complemented
7	#20243074	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×D₄×Dic₄₇	C₂²	1	4	yes	—
10	#15689235	C₄	1	16	yes	complemented
11	C₂×#13093956	C₂²	1	2	yes	—
12	C₂×#13093945	C₂²	1	4	yes	—
13	C₂×#13093935	C₂²	1	1	yes	characteristic
14	C₂×#13093934	C₂²	1	2	yes	—
15	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
16	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	C₂²⋊C₄×Dic₄₇	—	2	4	no	—
19	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
20	#15688969	C₂²	1	1	yes	characteristic
21	C₂×C₄₇⋊#95	—	2	2	no	—
22	#15688886	C₂²	1	2	yes	—
23	#15688885	C₂²	1	4	yes	—
24	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
26	C₄×D₈×C₄₇	C₄	1	8	yes	complemented
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂³×Dic₄₇	—	2	4	no	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	D₄×Dic₄₇	—	2	8	no	—
33	D₄×Dic₄₇	C₂×C₄	1	16	yes	complemented
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	4	no	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	#13093956	C₂×C₄	1	8	yes	complemented
40	#13093945	C₂×C₄	1	16	yes	complemented
41	#13093935	—	2	2	no	—
42	#13093934	C₂×C₄	1	8	yes	complemented
43	C₈×Dic₄₇	—	2	4	no	—
44	C₄₇⋊#95	—	2	2	no	—
45	C₄₇⋊#95	—	2	4	no	—
46	C₄₇⋊#95	D₄	1	2	yes	—
47	C₄₇⋊#95	D₄	1	2	yes	—
48	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₄₇	—	2	2	no	—
50	C₂×D₈×C₄₇	C₂×C₄	1	8	yes	—
51	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₄×D₄×C₄₇	—	2	4	no	—
55	C₄×D₄×C₄₇	C₂×C₄	1	8	yes	complemented
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	#107×C₄₇	C₂×C₄	1	4	yes	complemented
60	#102×C₄₇	C₂×C₄	1	8	yes	complemented
61	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
62	C₂⁴×C₄₇	—	2	2	no	—
63	C₂×D₄×C₄₇	—	2	2	no	—
64	C₂×D₄×C₄₇	—	2	4	no	—
65	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
66	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	8	no	—
69	C₂²×C₄×C₄₇	—	8	4	no	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	2	16	no	—
75	C₂²×Dic₄₇	—	8	4	no	—
76	C₂²×Dic₄₇	—	8	4	no	—
77	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
78	D₈×C₄₇	C₄²	1	16	yes	complemented
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₄₇	—	2	2	no	—
84	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×C₄₇	—	2	8	no	—
86	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₇	#41	1	2	yes	—
88	C₂²⋊Dic₄₇	—	2	16	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
91	C₄×Dic₄₇	—	8	4	no	—
92	C₄×Dic₄₇	—	8	2	no	—
93	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
94	C₄×Dic₄₇	#41	1	4	yes	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
98	C₂³×C₄₇	—	2	8	no	—
99	C₂³×C₄₇	—	2	4	no	—
100	C₂³×C₄₇	—	4	4	no	—
101	C₂³×C₄₇	—	4	2	no	—
102	C₂³×C₄₇	—	4	2	no	—
103	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₄₇	—	2	8	no	—
105	D₄×C₄₇	—	2	4	no	—
106	D₄×C₄₇	C₂×C₄²	1	8	yes	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	8	8	no	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	8	16	no	—
124	C₂×Dic₄₇	C₄×D₄	1	8	yes	—
125	C₈×C₄₇	—	2	2	no	—
126	C₈×C₄₇	C₂×C₄²	1	4	yes	—
127	C₄₇⋊C₈	—	2	4	no	—
128	#20503	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	8	no	—
130	C₂²×C₄₇	—	4	8	no	—
131	C₂²×C₄₇	—	4	8	no	—
132	C₂²×C₄₇	—	4	4	no	—
133	C₂²×C₄₇	—	4	4	no	—
134	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
135	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₄₇	#576	1	2	yes	—
137	C₂²×C₄₇	#577	1	2	yes	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	4	4	no	—
141	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₄₇	C₄×#41	1	2	yes	—
143	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
144	Dic₄₇	—	2	8	no	—
145	Dic₄₇	—	4	8	no	—
146	C₂×#442	—	47	2	no	—
147	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
148	#2275	—	47	2	no	—
149	#1776	—	47	2	no	—
150	C₄×#106	—	47	1	no	—
151	#1762	—	47	2	no	—
152	C₄×#102	—	47	4	no	—
153	#1726	—	47	1	no	—
154	C₂×C₄₇	—	4	8	no	—
155	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
156	C₂×C₄₇	#2946	1	4	yes	—
157	C₂×C₄₇	#2948	1	1	yes	characteristic
158	C₂×C₄₇	#2952	1	1	yes	characteristic
159	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
160	C₂×C₄×D₄	—	47	4	no	—
161	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
162	#442	—	47	16	no	—
163	C₄×D₈	Dic₄₇	1	8	yes	complemented
164	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
165	C₂×#106	—	47	2	no	—
166	C₂×#102	—	47	4	no	—
167	C₂×#102	C₂×D₄₇	1	2	yes	—
168	C₂×#97	—	47	2	no	—
169	C₂×#97	—	47	1	no	—
170	C₂×C₄×C₈	—	47	2	no	—
171	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
172	C₄×C₄⋊C₄	—	47	4	no	—
173	C₄×C₄⋊C₄	—	47	2	no	—
174	C₄×C₂²⋊C₄	—	94	4	no	—
175	C₄×C₂²⋊C₄	—	94	2	no	—
176	#376	—	47	1	no	—
177	C₂×#95	—	94	2	no	—
178	#340	—	47	2	no	—
179	#340	—	47	4	no	—
180	C₄₇	#20503	1	1	yes	characteristic, complemented, Sylow
181	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	—	94	4	no	—
184	C₂×D₈	C₄×D₄₇	1	8	yes	—
185	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
189	C₄×D₄	—	2	4	no	—
190	C₄×D₄	—	47	16	no	—
191	C₄×D₄	—	94	8	no	—
192	C₄×D₄	C₂×Dic₄₇	1	8	yes	complemented
193	C₂×C₄⋊C₄	—	47	4	no	—
194	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	94	4	no	—
197	C₂×C₄²	—	47	2	no	—
198	C₂×C₄²	—	94	4	no	—
199	C₂×C₄²	—	94	2	no	—
200	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
201	#107	C₂×Dic₄₇	1	4	yes	complemented
202	#106	—	47	8	no	—
203	#102	—	47	16	no	—
204	#102	C₂×Dic₄₇	1	8	yes	complemented
205	#97	—	47	8	no	—
206	#97	—	94	2	no	—
207	C₄×C₈	—	94	4	no	—
208	C₄×C₈	C₂×Dic₄₇	1	4	yes	complemented
209	#95	—	47	2	no	—
210	#95	—	47	2	no	—
211	#95	—	94	4	no	—
212	#95	—	94	2	no	—
213	C₂⁴	—	2	2	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	C₂×C₄×D₄₇	1	8	yes	—
217	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	8	4	no	—
221	C₂²×C₄	—	47	4	no	—
222	C₂²×C₄	—	94	16	no	—
223	C₂²×C₄	—	94	4	no	—
224	C₂²×C₄	—	376	4	no	—
225	C₂²×C₄	—	376	4	no	—
226	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	D₈	C₄×Dic₄₇	1	16	yes	complemented
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	47	4	no	—
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	—	94	2	no	—
234	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
235	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
236	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	47	8	no	—
239	C₄⋊C₄	—	94	4	no	—
240	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	94	16	no	—
243	C₄²	—	47	4	no	—
244	C₄²	—	47	4	no	—
245	C₄²	—	376	2	no	—
246	C₄²	—	376	4	no	—
247	C₄²	#10625352	1	2	yes	—
248	C₄²	C₂²×Dic₄₇	1	2	yes	—
249	C₂³	—	2	8	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	4	4	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
255	D₄	—	2	8	no	—
256	D₄	—	2	4	no	—
257	D₄	C₂×C₄×Dic₄₇	1	8	yes	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	8	8	no	—
263	C₂×C₄	—	47	8	no	—
264	C₂×C₄	—	94	4	no	—
265	C₂×C₄	—	94	4	no	—
266	C₂×C₄	—	94	8	no	—
267	C₂×C₄	—	376	16	no	—
268	C₂×C₄	—	188	4	no	—
269	C₂×C₄	—	188	4	no	—
270	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
271	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
273	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
274	C₂×C₄	#41×D₄₇	1	2	yes	—
275	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	94	4	no	—
278	C₈	C₂×C₄×Dic₄₇	1	4	yes	—
279	C₂²	—	2	8	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	8	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	C₄×D₄×D₄₇	1	2	yes	—
285	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
286	C₂²	#15689924	1	2	yes	—
287	C₂²	#15689930	1	2	yes	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	94	8	no	—
292	C₄	—	188	8	no	—
293	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
294	C₄	C₄×#10625352	1	2	yes	—
295	C₄	#41×Dic₄₇	1	1	yes	characteristic
296	C₂	—	4	8	no	—
297	C₂	C₄×D₄×Dic₄₇	1	1	yes	characteristic
298	C₂	#20252213	1	4	yes	—
299	C₂	#20252240	1	1	yes	characteristic
300	C₂	#20252272	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	#20503
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #20245749	C₂	semidirect
3. #20243287	C₂	semidirect
4. C₄×#13093956	C₂	semidirect
5. #20243213	C₂	semidirect
6. #102×Dic₄₇	C₂	semidirect
7. #20243074	C₂	semidirect
10. #15689235	C₄	semidirect
26. C₄×D₈×C₄₇	C₄	semidirect
33. D₄×Dic₄₇	C₂×C₄	semidirect
39. #13093956	C₂×C₄	semidirect
40. #13093945	C₂×C₄	semidirect
42. #13093934	C₂×C₄	semidirect
55. C₄×D₄×C₄₇	C₂×C₄	semidirect
59. #107×C₄₇	C₂×C₄	semidirect
60. #102×C₄₇	C₂×C₄	semidirect
61. C₄×C₈×C₄₇	C₂×C₄	semidirect
78. D₈×C₄₇	C₄²	semidirect
163. C₄×D₈	Dic₄₇	semidirect
180. C₄₇	#20503	semidirect
192. C₄×D₄	C₂×Dic₄₇	semidirect
201. #107	C₂×Dic₄₇	semidirect
204. #102	C₂×Dic₄₇	semidirect
208. C₄×C₈	C₂×Dic₄₇	semidirect
229. D₈	C₄×Dic₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15689235	2
#20245749	2
#20243287	2
C₄×#13093956	2
#20243213	2
#102×Dic₄₇	2
#20243074	2
C₂×C₄×D₈×C₄₇	2
#20503	47